Lefschetz–Verdier Trace Formula and a Generalization of a Theorem of Fujiwara

The goal of this paper is to generalize a theorem of Fujiwara (Deligne’s conjecture) to the situation appearing in a joint work [KV] with David Kazhdan on the global Langlands correspondence over function fields. Moreover, our proof is more elementary than the original one and stays in the realm of ordinary algebraic geometry, that is, does not use rigid geometry. We also give a proof of the Lefschetz–Verdier trace formula and of the additivity of filtered trace maps, thus making the paper essentially self-contained. The work was supported by the Israel Science Foundation (Grant No. 555/04) Received: May 2005 Accepted: August 2005     

Oxidation of the [Rh(CO)2Cl2] anion in aqueous solution

The reaction of rhodium(I) carbonyl chloride, [Rh(CO)₂Cl]₂, with dichromate, cerium(IV) sulfate, hexachloroplatinic acid or p-benzoquinone in aqueous hydrochloric acid proceeds by consumption of 4 equivalents of oxidizing agent per mole or rhodium(I) in accordance with the equation Rh^I(CO)₂  4e + H₂O → Rh^III(CO) + 2H⁺ + CO₂A “cyclic” oxidation mechanism is suggested.     

Comparative studies on photo-ionization and electron-impact ionization of peptide derivatives

Photo-ionization (PI) mass spectrometry performed with a monochromatic photon beam was applied to a series of peptide derivatives. PI mass spectra of ten N-acylpeptide methyl esters containing two to four residues of glycine, alanine, valine, leucine, proline, tryptophan, tyrosine, phenylalanine, methionine, carboxymethylcysteine, lysine and ornithine were studied. Comparative analysis of PI (10.2 eV) and electron-impact ionization (EI) (70 eV) mass spectra shows the total number of peaks on PI to be much less than that obtained with EI, especially in the low m/e region (< 250 to 300). At the same time the relative abundance of ‘heavy’ ions, including molecular ions, is much higher in PI. The amino acid fragmentation pattern followed by N-acylpeptide esters in PI was found to be the same as for EI.     

Photoionisation mass spectra of volatile derivatives of short peptides and appearance potentials of their characteristic ions

Mass spectra of the volatile derivatives of short peptides were studied by the photoionisation method with the use of monochromatic photons. The dependence of the intensity of the main peaks on the photon energy was analysed from 7·5 to 13·0 eV. The data obtain reveal the influence of the chemical structure of amino acid residues on the relative probability of the decomposition of peptide molecular ions at the CH? CO and CO? NH bonds, resulting in the formation of positively charged aldimine and amino acid N-terminal fragments, respectively. These data may be used as a basis for the application of the photoionisation technique to mass spectrometric sequential studies in peptides. In peptides containing residues of aliphatic amino acid the decomposition results mainly in the formation of aldimine ions, the stability of which increase with the increase of the alkyl chain size. In peptides containing residues of aromatic amino acids the decomposition is usually observed leading to formation of the amino acid ions. Ionisation potentials, as well as photoionisation efficiency curves and appearance potentials were determined for characteristic ions. Comparison was made of the values of the appearance potentials of different fragments formed upon decomposition of molecular ions. It has been shown that for peptides containing aliphatic amino acid moieties the appearance potentials of aldimine fragments are always lower than those inherent to peptides containing aromatic amino acid residues. The larger the size of an aliphatic chain, the lower is the energy of formation of these fragments. For all the compounds studied, including the peptides containing aromatic amino acid residues, the appearance potentials of the aldimine ions did not exceed those of the amino acid ions. These data indicate that, contrary to the experiments with electron-impact with energy of about 70 eV, upon ionisation with photons with energy from 7·5 to 13·0 eV, the aldimine fragments appear directly due to primary decomposition of molecular ions, independent of the formation of the amino acid fragments. The photoionisation efficiency curves for peptides containing different types of amino acid residues facilitate the choice of an optimal photon energy providing unequivocal information on the amino acid sequence in the peptide under study.     

Geometric approach to parabolic induction

In this paper we construct a “restriction” map from the cocenter of a reductive group G over a local non-archimedean field F to the cocenter of a Levi subgroup. We show that the dual map corresponds to parabolic induction and deduce that parabolic induction preserves stability. We also give a new (purely geometric) proof that the character of normalized parabolic induction does not depend on the parabolic subgroup. In the appendix, we use a similar argument to extend a theorem of Lusztig–Spaltenstein on induced unipotent classes to all infinite fields. We also prove a group version of a theorem of Harish-Chandra about the density of the span of regular semisimple orbital integrals.     

Interfragment transmission of electronic effects of π-acceptor ligands in heterometallic triad complexes with trans-[Ru(Py)4(CN)2] as a central unit

The complex trans-[RuPy₄(CN)₂] cleaves chloride bridges in the binuclear rhodium(i) and palladium(ii) complexes [Rh(CO)₂Cl]₂, [Rh(η⁴-C₈H₁₂)Cl]₂, [(η⁴-C₈H₁₂)Rh(μ-Cl)₂Rh(CO)₂], [Pd(η³-C₃H₅)Cl]₂, and [(η₃-C₃H₅)Pd(μ-Cl)₂Rh(CO)₂] to form heterometallic triad complexes [(CO)₂ClRh(NC)RuPy₄(CN)RhCl(CO)₂] (1), [(η⁴-C₈H₁₂)ClRh(NC)RuPy₄(CN)RhCl-(η⁴-C₈H₁₂)] (2), [(CO)₂ClRh(NC)RuPy₄(CN)RhCl(η⁴-C₈H₁₂)] (3), [(η³-C₃H₅)ClPd(NC)-Ru(Py)₄(CN)PdCl(η³-C₃H₅)] (4), and [(CO)₂ClRh(NC)Ru(Py)₄(CN)PdCl(η₃-C₃H₅)] (5), respectively. In solutions, complex 3 coexists with equilibrium amounts of compounds 1 and 2; complex 5 is in the equilibrium with compounds 4 and 1. In both cases, the ratio of concentrations is close to binomial. Complexes 2 and 5 treated with [Rh(CO)₂Cl]₂ are converted into 1 with the simultaneous formation of [Rh(η⁴-C₈H₁₂)Cl]₂ and [Pd(η³-C₃H₅)Cl]₂, respectively. The δ_H and δ_C values for the ligands η⁴-C₈H₁₂, η³-C₃H₅, and CO are sensitive to the nature of the remote triad unit. The ligand effects are shown to be transmitted along the chain L′-M′-(NC)-Ru-(CN)-M″-L″.     

Mass spectrometric studies on the effect of sidechains of N-acetylamino acid methyl esters and N-methylamides on their photoionisation

Photoionisation mass spectra of N-methylamides of N-acetyl derivatives of the following amino acids have been studied: alanine, valine, leucine, serine, threonine, phenylalanine, tyrosine, tryptophan and proline. The study also included methyl esters of N-acetyl tyrosine, N-acetyl phenyl alanine and N-trideuteroacetyl phenylalanine. The photon energies were within the range of 7.5 to 13.0 eV. The appearance potentials of the main ions were also determined. The compounds under study contain CH? CO, CO? NH and N? C_α bonds as well as sidechains and may therefore be regarded as models of short peptides. Under photoionisation conditions, decomposition of compounds containing aliphatic sidechains involves mainly CH? CO bonds; the appearance potentials of the corresponding fragments decrease with an increase in the size of sidechain. The presence of hydroxyl in the sidechain results in an increase in the appearance potential of a fragment formed due to CH? CO bond rupture, whereas the presence of an aromatic moiety favours an even greater increase in the appearance potential and results in predominant rupture of the N? C_α bond. Analysis of the mass spectra obtained confirms the conclusion made previously that application of the photoionisation technique enables one to increase considerably the relative intensity of characteristic peaks and to suppress secondary process taking place upon ionisation by the electrons with energy values of the order of 70 eV.